Operations on Fuzzy Sets

Given ‘X’ to be universe of discourse, A and B are two fuzzy sets with membership function μA(x) and μB(x) then,

Union

The union of two fuzzy sets A and B is a new fuzzy set A ∪ B also on ‘X’ with membership function defined as follow:

Membership Fuction for Union of Fuzzy Sets

Intersection

Intersection of fuzzy sets A & B, is a new fuzzy set A ∩ B also on ‘X’ whose membership function is defined by

Intersection of Fuzzy Sets Membership Function

Compliment

Compliment of a fuzzy set A is A with membership function

Compliment of Fuzzy Set A Membership Function

Product of Two Fuzzy Sets

The product of two fuzzy sets A & B is a new fuzzy set A.B with membership function:

membership function for the product of fuzzy sets

Equality

Two fuzzy sets A and B are said to be equal i.e, A = B if and only if μA(x) = μB(x) Which means their membership values must be equal.

Product of Fuzzy Sets with a Crisp Number

Multiplying a fuzzy set A by a crisp number ‘n’ results in a new fuzzy set n.A, whose membership function is

Product of fuzzy set with a crisp number

Power of a Fuzzy Set

The alpha power of a fuzzy set A is a new fuzzy set Aα whose membership function is:

Power of a Fuzzy Set Membership Function

that is, individual memberships power of α

Difference of Fuzzy Sets

The differences of two fuzzy sets A and B is a new fuzzy set A-B which is defined as

Difference of Fuzzy Sets A & B

Disjunctive Sum of A & B

It is the new fuzzy set defined as follow:

Disjoint Sum of A and B

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